This could be written as cot (2x) = x + 1/x And for x> 0. x + 1/x > 1 And observing the nature of both the graphs we can see they intersect at countably infinite points .
My attempt: We are trying to find zeroes of the function f(x)=(1+x^2)*tan(2x) - x. f(pi/4 +) = - infiinty. f(3pi/4 -) = + infinity. So there must be a root of f in the interval (pi/4, 3pi/4) using intermediate value property of f on (pi/4, 3pi/4). I am not sure how to show that there is exactly one root in (pi/4, 3pi/4). I think requires the results on differentiability...
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This could be written as cot (2x) = x + 1/x
And for x> 0. x + 1/x > 1
And observing the nature of both the graphs we can see they intersect at countably infinite points .
What is the idea behind assuming this function sir!
My attempt:
We are trying to find zeroes of the function f(x)=(1+x^2)*tan(2x) - x.
f(pi/4 +) = - infiinty.
f(3pi/4 -) = + infinity.
So there must be a root of f in the interval (pi/4, 3pi/4) using intermediate value property of f on (pi/4, 3pi/4).
I am not sure how to show that there is exactly one root in (pi/4, 3pi/4).
I think requires the results on differentiability...